The Basic Linear-Quadratic Model

1. Introduction to the Linear-Quadratic Model

The Linear-Quadratic (LQ) model describes how cells respond to radiation based on the total dose delivered and the dose rate. The model is used to predict cell survival following exposure to a given dose of radiation, particularly in the context of fractionated radiation therapy.

The basic equation for the survival fraction \( S \) following an acute dose \( d \) is given by:

\( S(d) = \exp\left( -\alpha d - \beta d^2 \right) \)

Where:

α (alpha): The linear sensitivity coefficient (units of Gy^–1), which describes the damage caused by single hits (linear damage).
β (beta): The quadratic sensitivity coefficient (units of Gy^–2), which accounts for the damage caused by two-hit processes (e.g., double strand breaks).

2. Survival Following Fractionated Treatment

If the treatment is repeated in \( N \) well-spaced fractions, the net survival is \( S_N \). The equation for survival after \( N \) fractions is:

\( S_N = \exp\left( -\alpha N d - \beta N d^2 \right) \)

Taking the natural logarithm of both sides of this equation, and dividing throughout by \( \alpha \), we get:

\( \frac{\ln(S_N)}{\alpha} = -N d - \frac{\beta N d^2}{\alpha} \)

This form of the equation is useful for calculating survival fractions in fractionated treatments, where the total dose is split into smaller doses delivered over time.

3. Interpretation and Application

The LQ model helps in understanding how cells respond to radiation depending on the fractionation scheme (how the total dose is divided over multiple sessions). It accounts for:

The damage caused by low-dose radiation (linear component, proportional to dose).
The damage caused by higher-dose radiation, where two independent hits increase the likelihood of lethal damage (quadratic component, proportional to dose²).

The linear and quadratic coefficients, \( \alpha \) and \( \beta \), can vary for different cell types, tissues, and even the radiation quality (LET), affecting the shape of the survival curve.

Note on Fractionation:

The concept of fractionated radiation is critical in clinical settings. By spreading the dose over multiple treatments, normal tissues can repair some of the damage between sessions, while the tumor cells, which are less efficient at repair, accumulate damage more quickly. This principle is a core part of radiation therapy strategies for cancer treatment.

Example: Using the LQ Model in Therapy

In clinical practice, the LQ model is used to determine the optimal fractionation scheme for a given tumor type. For example, the choice of whether to deliver 2 Gy per fraction (standard fractionation) or 3 Gy per fraction (hypofractionation) depends on the sensitivity of the tumor and the surrounding healthy tissues, which can be predicted using the LQ model.